This invention relates to waveguides, and more particularly to designing curved waveguides so as to minimize bend-induced transmission losses.
A xe2x80x9cwaveguidexe2x80x9d can be generally defined as a structure that transmits electromagnetic waves from one point to another. As compared to transmission in an unbounded medium, transmission in a waveguide limits wave intensity to a finite cross section and may guide the wave along a path that is not straight. A waveguide may be a parallel plate type waveguide, or may be any rectangular or circular xe2x80x9cpipexe2x80x9d that confines and guides electromagnetic waves between two locations.
Waveguides may be designed to carry waves of any wavelength, commonly, radio frequency or optical frequency waves. For optical frequencies, reflection off an interface from an optically dense medium to one that is less dense provides a means to guide waves. Dielectric waveguides in integrated circuitry and optical fibers have found important applications for optical frequencies.
Practically all transmission, communication, and sensor systems that use waveguides have one or more bends in the waveguides. The bends are present to accommodate the systems to particular geometries, to reduce size, or because of the requirements imposed by the physics of operation. As an example of the latter case, a Sagnac interferometer requires enclosure of an area by the waveguide.
In general, curvature in a waveguide introduces intrinsic losses due to radiation. Intuitively, this can be understood because any attempt to force a wavefront to travel around a curve results in phase velocities at large radii that exceed the velocity of light. This results in the imposition of a radial component to the direction of propagation of the wavefront at large radii, i.e. there will be outward radiation. These losses set limits on system sensitivity for a given electromagnetic source, because the signal amplitude at a detector is decreased. It also places a lower limit on system size, because radiation losses increase as the radius of curvature decreases.
In conventional microwave waveguides that are completely enclosed by conducting boundaries, these xe2x80x9cbend-inducedxe2x80x9d radiation losses are prevented. However, the enclosure introduces additional transmission losses due to the eddy currents in the skin depth surface layer of the conductor. In optical waveguides, the conducting reflecting boundary is usually absent, and the radiation losses can be dominant.
Much of the reported work in modeling bend-induced radiation losses involves numerical approximations. The methods employed include Fourier decomposition, finite element analysis, point matching of amplitudes and derivatives at boundaries between different regions, matching of Fourier coefficients at boundaries, replacing derivatives by finite differences, beam propagation, variational approximations, reciprocity techniques, and various modal expansion and perturbation approaches.
One aspect of the invention is a curved waveguide for minimizing bend-induced radiation losses of a guided wave. The waveguide has three regions, a core region and two regions outside the core. The index of refraction of each region is calculated so that the electromagnetic field within the waveguide has certain characteristics. Specifically, in the core region, the field is trigonometric. In the intermediate region, the field is hypertrigonometric, i.e., evanescent. In the outer region, the field is trigonometric but with an amplitude smaller than in the core region. Further calculations are used to calculate the boundaries between adjacent regions, and hence the width of each region, so as to further minimize losses.
An advantage of the invention is that the curved waveguide can be designed to minimize losses. One design feature is the use of three regions of varying indices of refraction. A core, as well as two different regions outside the core, are used. For each region, the index of refraction and the width of the region is designed to xe2x80x9cshapexe2x80x9d the electromagnetic field within the waveguide such that electromagnetic energy is contained primarily in the core region of the waveguide. Additionally, a reflector in the outermost region, where the wave amplitude is very small, minimizes skin depth losses.